47

Slide Rule Scale Listing Conventions

David Sweetman

Introduction Types of Slide Rules

While doing the layout for the Journal, I noticed that article One of the problems with listing scales is the consideration authors provide many different ways of listing scales on of the different types of slide rules. Essentially, there are slide rules. Given that one of the functions of the Editor is three physical construction types, with some variations of to use proper grammar (as is appropriate for a professional each type. For the purposes of this article, the following will journal) and to standardize layout (formatting), both for ease be considered: of reading and consistency, I wondered what to do? After reviewing a previous Journal article1, then asking a number 1. Linear; the scales are nominally 12.5, 25, or 50 cm of frequent contributors, the following is a summary of long. Scales may be: many of the points addressed, identifying possible options, a. On the front of the stock and slide. and then a proposed solution. The reasons for choosing b. On the front and back2 of the stock and slide. among the options for the proposed solution are provided. c. On the front and back of the slide only. In general, the emphasis on layout is for reasons of d. In the “gutter” (well), i.e., on the stock under the standardization, i.e., for the benefit of the reader as well as slide. the Layout Editor. Authors may choose to use their own e. There may be constants or conversion factors or format for submission, but the Editor and Layout Editor formulas on the back of the stock. may revise as appropriate. Oughtred Society members are encouraged to submit their opinions and comments, 2. Circular; the scales are of a variety of lengths, preferably using the Oughtred Society Message Board depending on the diameter of the . May also (www.oughtred.org) or send to me [email protected]. use the rim or edge. Scales may be: a. On the front or front & back of a single fixed disk with one or more rotating cursors. b. On the front or front & back of a disk with two or more rotating rings and cursors. 48 Journal of the Oughtred Society

3. Cylindrical; the scales are of a variety of lengths,  How should notes, comments, or other methods to depending on the length and diameter of the slide rule. identify unique attributes be included? Scales may be: a. For a drum and bar (e.g., Thacher) on both the  How are layout concerns to be implemented, e.g., two- drum and bars. There may be scales on multiple columns with left/right justification, 10 TNR (Times surfaces of the bars. New Roman, size 10) font? b. For a two cylinder (e.g., Otis King) on both cylinders. Proposed Requirements

Options Given that whatever the final outcome, the solution will be a compromise, meaning that all contributing parties will When considering how to list scales, there are a number of NOT be satisfied with the solution. The following solution options, many with competing solutions. Although the that contains guidelines for authors and requirements for initial concern was with arranging the layout/format, the editors are what the Journal’s Editor thinks will best meet primary concern is the readability and ease of understanding the needs of the membership, as well as those of the editing for the reader. staff:

These options, in no particular order, include: 1. Scales should be identified using Oughtred Society standard symbols, as shown in Table 1.  Should the Journal adopt the format used in a book? Should the Journal adopt one of the formats used by 2. When listing for a specific slide rule that uses a symbol authors in various Journal articles? If so, which format? that is non-standard or no symbol, the listing should use the standard symbol, with a note identifying what  Should the Journal use symbols or words to identify symbol is actually on the slide rule. If comparing two physical, logical, or other interfaces? or more slide rules, the list could use the actual symbol for each slide rule, with a comment noting if there is a  What symbols should be used for the scales? Should a standard symbol for the scale. table of standard scale symbols for the Journal be generated? 3. If there is no standard symbol, then the list will include the symbol on the slide rule, with a note defining the  What if there is no symbol for a scale or a standard meaning of the symbol. symbol is not used? 4. Scale symbols should be listed in order from top-to-  What symbols or other conventions should be used to bottom for linear and outer-to-inner for circular slide identify or segregate the physical surfaces the scales are rules. Cylindrical slide rule scale lists should start with printed on? the most commonly used scale on a drum and bar slide rule or outer-to-inner for two-cylinder slide rules.  How should the relationship between scales and boundaries be identified, e.g., based on physical 5. The scales on the front (obverse) of the slide rule should structure of the slide rule, based on logical be listed before those on the back (reverse). Given that (mathematical) interaction of the scales? determining the front or back can be arbitrary, when necessary the author should state why a particular side  When listing, how should the scales be separated, e.g., is the front. a symbol, a punctuation mark, a fixed or variable space? 6. The scales on the stock (body, stator) or outer ring or bars should be separated from the scales on the slide or  How should scales that are unique to a certain slide rule inner ring or drum. be identified, e.g., special symbols, words? 7. The scales on different adjacent surfaces, e.g., beveled,  In what order should scales be listed, e.g., top-to- right-angle edges, should be separated from the other bottom, front-to-back, outer-to-inner? scales on the stock.

 How are rare scale placements, e.g., in the gutter of a 8. All other scale information, e.g., scales in the gutter linear slide rule, to be listed? (well) of the stock, should be added with comments, preferably as a separate list. Other information, e.g.,  How should differences between linear, circular, and constants or formulas on the back, may be included cylindrical slide rules be identified? using comments. Volume 24, Number 1, Spring 2015 49

While an attempt has been made to use a consistent 4. For a Pickett Model 9: methodology for linear, circular, and cylindrical slide rules, B24 B/C S d1 d2 d3/P PM there is a degree of variability between the three major A/B C/D L physical forms, as well as variation between manufacturers. This is a special slide rule with unique scales for the Therefore, the guidelines and requirements are not absolute, electricity transmission, so the above format is an but represent guiding principles for the desired intent. example of how to list non-standard scale symbols.

Proposed Solution Reasoning

The following guidelines would be used for layout The following is the reasoning for each suggested guideline. (formatting) of articles printed in the Journal: While the guidelines were developed primarily to address linear slide rules, the guidelines also apply to circular and 1. The scale symbols should be the same as those in Table cylindrical slide rules. Given that slide charts, whether 1. If not listed in Table 1, use the symbol or notation on linear or circular almost always have specialty scales, the slide rule. whether logarithmic or not, the listing of their scales is not a primary constraint. When members suggest different or 2. The list should start on a new line, i.e., not be embedded changed guidelines, the reasons need to be provided. in a sentence within a paragraph, indented if appropriate. Given that we are using lists, the 1. Different manufacturers in different countries used a introductory words should end with a colon (:). Very variety of symbols for the same function. Table 1 short lists, e.g., when identifying specific scales for attempts to list the most commonly accepted symbols reference, may be included in the text. for a given scale. Not all symbols or scales are listed, because for some scale names or symbols, there is just 3. The list should start from the top of the front side, e.g., too much variation or too many different a linear slide rule or the outer ring of a circular slide interpretations. These unusual scales should be rule. Drum & bar cylindrical slide rules will probably identified with the symbol printed on the slide rule, start at some arbitrary point on the external ribs, then with an accompanying note. If there is no symbol on list on the drum. Two-cylinder slide rules start with the the slide rule, the symbol that matches the scale outer cylinder. function should be used; otherwise use a generic symbol, e.g., §, with a notation that the meaning of the 4. A / separates the adjacent surfaces that can move symbol § is the description of the scale. relative to each other, e.g., stock and slide, inner and outer ring. There is no space on either side of the / 2. Starting the list on a new line is proper grammar symbol. (including punctuation) for a long list, as well as visually segregating the symbols (and their context) 5. A // separates scale(s) on a different adjacent surface, from the wording. Even for slide rules with few scales, e.g., a beveled or right-angle edge on a linear slide rule. the list is still considered long. There is an en space3 on each side of the // symbol. 3. Identifying the “top” is based on reading English, i.e., 6. When there are both front and back scale symbols, they one starts reading from the top left corner of the text. are listed on separate lines. When looking at the front (whether identified by the manufacturer, based on the number or frequency of use 7. When there are only scale symbols on the back of the of the scales, or other factor), the top is where one slide, they are listed on the same line in [ ]. would start reading the English words or symbols (and is the method of starting to read for most other 8. Each scale symbol is separated by an en space. languages used for slide rules).

Examples 4. The primary choice is identifying whether the symbol separating the moving components, e.g., the stock and 1. For a K&E N4081-3: slide or inner and outer ring or bar and drum, should be LL0 LL00 A/B T ST S/T DI K based on a physical description of the slide rule or a L LL1 DF/CF CIF CI C/D LL3 LL2 logical description of the relationship of the adjacent scales e.g., on the body and stock or inner and outer ring 2. For a K&E 4053-3 (see Figure 1): or bar and drum. The logical description seems to be in // A/B CI C/D K // cm [S L T] more prevalent and is more appropriate for the circular and cylindrical slide rules. 3. For a Nestler Darmstadt 21 (see Figure 2): cm L // K A/B CI C/D P // S T 5. The // symbol physically shows two surfaces with a [LL1 LL2 LL3] common edge, e.g., for the edges of linear slide rules or 50 Journal of the Oughtred Society

triangular rotatable bars of cylindrical slide rules. Other Acknowledgements symbols could be used, e.g., to indicate a beveled vs. a right-angle edge, but the added information is not worth This article would not have been possible without the the additional complexity for defining construction substantial contributions of David Rance, Rodger Shepherd, symbols. Clark McCoy, Otto van Poelje, Marion Moon, Clay Castleberry, Tom Wyman, and Ed Chamberlain. 6. Within the Journal, fitting both the front and back scale symbols on a single line will often not be possible. Table 1 uses information from Slide Rules Their History, Putting the front and back scales on separate lines, Models, and Makers by Peter Hopp, Slide Rules (A.W. clearly separates the two major surfaces. Faber, A.W. Faber-Castell) by Peter Holland, and The Oughtred Society Reference Manual by Ted Hume and Bob 7. There are linear slide rules that do not have scales on Koppany. the back of the stock and do not have many scales on the front, but there are two or three scales on the back of the slide. Use the [ ] to identify the scales on the back of slide, still on the same line with the other scales to minimize the space required.

8. How to separate the scale symbols was by far the most difficult decision, with the most options. The major consideration is the impact of the separation methodology on both the visual presentation and the physical layout. Using another symbol or punctuation mark would not add any pertinent information, while still leaving the issue of how to handle spacing. Not having a space after a punctuation mark is bad grammar and visually unappealing. Having a variable space (the standard space bar proportional spacing) would change FIGURE 1. Image of K&E 4053-3 Actual Scales the spacing on each line because of the left/right justification of the Journal contents. There are several fixed space intervals available; however, the en3 space seems to be the best compromise of a fixed physical separation and a clear separation of scale symbols.

FIGURE 2. Image of Nestler 21 Actual Scales Notes

1. Hudson, Don, Comments on Slide Rule Nomenclature, Journal of the Oughtred Society, 1:2, August 1992, pages 5-10. 2. What is the front or back is often not easy or obvious to differentiate for some slide rules. 3. The en space, or em space, are listed in the Special Characters section of the Insert Symbols menu in Word. http://en.wikipedia.org/wiki/En_%28typography%29 Volume 24, Number 1, Spring 2015 51

TABLE 1. Slide Rule Scale Symbols and Meanings

Symbol Mathematical Range Description Function Relationship

A x2 1 - 100 2-cycle, Square of D, on stock Find Squares & Square Roots B x2 1 - 100 2-cycle, Square of D, on slide Find Squares & Square Roots AI 1/x2 1 - 0.01 2-cycle, Reciprocal Square of A Reciprocals, Chain & Division BI 1/x2 1 - 0.01 2-cycle, Reciprocal Square of B Reciprocals, Chain Multiplication & Division C x 1 - 10 1-cycle, Principal scale, on stock Basic Multiplication & Division D x 1 - 10 1-cycle, Principal scale, on slide Basic Multiplication & Division CF πx 3.1 - 31 1-cycle, Folded at π or √10 Multiplies result by π CI 1/x 1 - 0.1 1-cycle, Reciprocal of C Reciprocals, Chain Multiplication & Division CIF 1/πx 0.33 – 0.03 1-cycle, Folded at π, Reciprocal of CF Divides result by π DF πx 3.1 - 31 1-cycle, Folded at π or √10 Multiplies result by π DI 1/x 1 - 0.1 1-cycle, Reciprocal of D Reciprocals, Chain Multiplication & Division DIF 1/πx 0.33 – 0.03 1-cycle, Folded at π, Reciprocal of DF Divides result by π K x3 1 - 1000 3-cycle, Cube of D Find Cubes & Cube Roots L log10(x) 0 - 1 Linear scale Mantissa of base10 logarithm of D LL, Ln ln(x) 0 – 2.7 Lower log-log scales Raise D to any power or extract any root LL0 e0.001x 1.001 – 1.01 Exponent of x·103 Log Log scales LL1 e0.01x 1.01 – 1.11 Exponent of x·102 Log Log scales LL2 e0.1x 1.1 – 3.0 Exponent of x·101 Log Log scales LL3 ex 2.5 – 100,000 Exponent of x Log Log scales LL00 e-0.001x 0.999 – 0.990 Exponent of -x·103 Reciprocal Log Log scales LL01 e-0.01x 0.99 – 0.90 Exponent of -x·102 Reciprocal Log Log scales LL02 e-0.1x 0.91 – 0.35 Exponent of -x·101 Reciprocal Log Log scales LL03 e-x 0.4 – 0.00001 Exponent of -x Reciprocal Log Log scales R1 √x 1 – 3.2 R1 is the of D Double Length Square Root Scale R2 √10x 3 - 10 R2 is the square root of 10D Double Length Square Root Scale P √(1-(0.1x)2) 0.996 - 0 Cosine of sin-1(D) Pythagorean scale S sin-1(x) 5.8° - 90° D is the sine of angle S Trigonometric sine T tan-1(x), cot-1(x) 5.8° - 90° D is the /cotangent of angle T Trigonometric tangent & cotangent T1 tan-1(x), cot-1(x) 5°-49°/41°-85° D is the tangent/cotangent of angle T Trigonometric tangent & cotangent T2 tan-1(x), cot-1(x) 41°-85°/5°-49° D is the tangent/cotangent of angle T Trigonometric tangent & cotangent ST sin-1(x), tan-1(x) 0.58° – 5.8° D is the sin or tan of angle ST Trigonometric sine or tangent for small angles Ch coth-1(x) 0.1 - 3 D is the hyperbolic cotangent of Ch Hyperbolic Cotangent Th tanh-1(x) 0.1 - 3 D is the hyperbolic tangent of Th Hyperbolic Tangent Sh1 sinh-1(x) 0.1 – 0.9 Hyperbolic sine of angle S For small values Sh2 sinh-1(x) 0.85 – 3 Hyperbolic sine of angle S For larger values H1 √(1+(0.1x)2) 1.005 - 1.5 cosh of sinh-1 of 0.1D Hypotenuse Scale H2 √(1+x2) 1.4 - 10 cosh of sinh-1 of D Hypotenuse Scale V Volts 0.5 – 10 Voltage drop For motors & generators W % 20 - 100 Efficiency For motors & generators § Arbitrary symbol For unlabeled scales with no standard symbol in Ruler in inches Convenience cm Ruler in centimeters Convenience